arXiv subjects

Petr Vojtěchovský

Publications and source records attributed to Petr Vojtěchovský.

At least 19 recordsLinked to original sources

arXiv2026

On the Parallel Complexity of Identifying Groups and Quasigroups via Decompositions

In this paper, we investigate the computational complexity of isomorphism testing for finite groups and quasigroups, given by their multiplication tables. We crucially take advantage of their various decompositions to show the following: - We first consider the class of groups that admit direct product decompositions, where each indecompsable factor is $O(1)$-generated, and either perfect or centerless. We show any group in this class is identified by the $O(1)$-dimensional count-free Weisfeiler--Leman (WL) algorithm with $O(\log \log n)$ rounds, and the $O(1)$-dimensional counting WL algorithm with $O(1)$ rounds. Consequently, the isomorphism problem for this class is in $\textsf{L}$. This improves upon the previous upper bound of $\textsf{TC}^{1}$, which was obtained using $O(\log n)$ rounds of the $O(1)$-dimensional counting WL (Grochow and Levet; FCT 2023, \textit{J. Comput. Syst. Sci.} 2026). - We next consider more generally, the class of groups where each indecomposable factor is $O(1)$-generated. We exhibit an $\textsf{AC}^{3}$ canonical labeling procedure for this class. Here, we accomplish this by showing that in the multiplication table model, the direct product decomposition can be computed in $\textsf{AC}^{3}$, parallelizing the work of Kayal and Nezhmetdinov (ICALP 2009). - Isomorphism testing between a central quasigroup $G$ and an arbitrary quasigroup $H$ is in $\textsf{NC}$. Here, we take advantage of the fact that central quasigroups admit an affine decomposition in terms of an underlying Abelian group. Only the trivial bound of $n^{\log(n)+O(1)}$-time was previously known for isomorphism testing of central quasigroups.

cs.DS↗

arXiv2025

Bol loops of order 27

We classify Bol loops of order $27$, using a combination of theoretical results and computer search. There are $15$ Bol loops of order $27$, including five groups. New constructions for the ten nonassociative Bol loops of order $27$ are given.

math.GR↗

arXiv2025

Algebraic invariants of multi-virtual links

Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves, we discuss the equivalence of multi-virtual link diagrams, particularly those that have the same virtual projections. We introduce operator quandles (that is, quandles with a list of pairwise commuting automorphisms) and construct an infinite family of connected operator quandles in which at least one third of right translations are distinct and pairwise commute. Using our set of generating moves, we establish the operator quandle coloring invariant and the operator quandle $2$-cocycle invariant for multi-virtual links, generalizing the well-known invariants for classical links. With these invariants at hand, we then classify certain small multi-virtual knots based on the existing tables of small virtual knots due to Bar-Natan and Green. Finally, to emphasize a key difference between virtual and multi-virtual knots, we construct an infinite family of pairwise nonequivalent multi-virtual knots, each with a single classical crossing. Many open problems are presented throughout the paper.

math.GT↗

arXiv2025

SAT-Based Techniques for Lexicographically Smallest Finite Models

This paper proposes SAT-based techniques to calculate a specific normal form of a given finite mathematical structure (model). The normal form is obtained by permuting the domain elements so that the representation of the structure is lexicographically smallest possible. Such a normal form is of interest to mathematicians as it enables easy cataloging of algebraic structures. In particular, two structures are isomorphic precisely when their normal forms are the same. This form is also natural to inspect as mathematicians have been using it routinely for many decades. We develop a novel approach where a SAT solver is used in a black-box fashion to compute the smallest representative. The approach constructs the representative gradually and searches the space of possible isomorphisms, requiring a small number of variables. However, the approach may lead to a large number of SAT calls and therefore we devise propagation techniques to reduce this number. The paper focuses on finite structures with a single binary operation (encompassing groups, semigroups, etc.). However, the approach is generalizable to arbitrary finite structures. We provide an implementation of the proposed algorithm and evaluate it on a variety of algebraic structures.

cs.LO↗

arXiv2023

Supernilpotent groups and $3$-supernilpotent loops

We find a short equational basis for the variety of $3$-supernilpotent loops. We also present a conceptually simple proof that $k$-nilpotence and $k$-supernilpotence are equivalent for groups. Connections between $3$-supernilpotent loops, Moufang loops, code loops, automorphic loops and AIM loops are explored.

math.GR↗

arXiv2023

Abelian congruences and solvability in Moufang loops

In groups, an abelian normal subgroup induces an abelian congruence. We construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop that does not induce an abelian congruence. On the other hand, we prove that in $6$-divisible Moufang loops, every abelian normal subloop induces an abelian congruence. In loops, congruence solvability adopted from the universal-algebraic commutator theory of congruence modular varieties is strictly stronger than classical solvability adopted from group theory. It is an open problem whether the two notions of solvability coincide in Moufang loops. We prove that they coincide in $6$-divisible Moufang loops and in Moufang loops of odd order. In fact, we show that every Moufang loop of odd order is congruence solvable, thus strengthening Glauberman's Odd Order Theorem for Moufang loops.

math.GR↗

arXiv2023

Congruence solvability in finite Moufang loops of order coprime to three

We prove that a normal subloop $X$ of a Moufang loop $Q$ induces an abelian congruence of $Q$ if and only if each inner mapping of $Q$ restricts to an automorphism of $X$ and $u(xy) = (uy)x$ for all $x,y\in X$ and $u\in Q$. The former condition can be omitted when $X$ is $3$-divisible. This characterization is then used to show that classically solvable finite $3$-divisible Moufang loops are congruence solvable.

math.GR↗

arXiv2023

On abelian-by-cyclic Moufang loops

We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if $Q$ is a group.

math.GR↗

arXiv2022

Estimating the Jacobian matrix of an unknown multivariate function from sample values by means of a neural network

We describe, implement and test a novel method for training neural networks to estimate the Jacobian matrix $J$ of an unknown multivariate function $F$. The training set is constructed from finitely many pairs $(x,F(x))$ and it contains no explicit information about $J$. The loss function for backpropagation is based on linear approximations and on a nearest neighbor search in the sample data. We formally establish an upper bound on the uniform norm of the error, in operator norm, between the estimated Jacobian matrix provided by the algorithm and the actual Jacobian matrix, under natural assumptions on the function, on the training set and on the loss of the neural network during training. The Jacobian matrix of a multivariate function contains a wealth of information about the function and it has numerous applications in science and engineering. The method given here represents a step in moving from black-box approximations of functions by neural networks to approximations that provide some structural information about the function in question.

cs.LG↗

arXiv2021

A Sufficient Condition for a Quandle to be Latin

A quandle is an algebraic structure satisfying three axioms: idempotency, right-invertibility and right self-distributivity. In quandles, right translations are permutations. The profile of a quandle is the list of cycle structures, one per right translation in the quandle. In this note we prove that if, for each cycle structure in the profile of a quandle, no two cycle lengths are equal, then the quandle is latin -- this is the sufficient condition mentioned in the title.

math.CO↗

arXiv2021

Set-theoretic Yang-Baxter (co)homology theory of involutive non-degenerate solutions

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang-Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang-Baxter equation and prove that the set-theoretic Yang-Baxter homology of certain solutions can be split into the normalized and degenerated parts.

math.GT↗

arXiv2021

A variety of Steiner loops satisfying Moufang's theorem: A solution to Rajah's Problem

A loop $X$ is said to satisfy Moufang's theorem if for every $x,y,z\in X$ such that $x(yz)=(xy)z$ the subloop generated by $x$, $y$, $z$ is a group. We prove that the variety $V$ of Steiner loops satisfying the identity $(xz)(((xy)z)(yz)) = ((xz)((xy)z))(yz)$ is not contained in the variety of Moufang loops, yet every loop in $V$ satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah.

math.GR↗

arXiv2021

Division sudokus: Invariants, enumeration and multiple partitions

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions.

math.CO↗

arXiv2020

Subdirect products and propagating equations with an application to Moufang theorem

We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops. An equation $\varepsilon$ propagates in an algebra $X$ if $\varepsilon(\overrightarrow y)$ holds whenever $\varepsilon(\overrightarrow x)$ holds and the elements of $\overrightarrow y$ are contained in the subalgebra of $X$ generated by $\overrightarrow x$. If $\varepsilon$ propagates in $X$ then it propagates in all subalgebras and products of $X$ but not necessarily in all homomorphic images of $X$. If $\mathcal V$ is a variety, the propagating core $\mathcal V_{[\varepsilon]} = \{X\in\mathcal V:\varepsilon$ propagates in $X\}$ is a quasivariety but not necessarily a variety. We prove by elementary means Goursat's Lemma for loops and describe all subdirect products of $X^k$ and all finitely generated loops in $\mathbf{HSP}(X)$ for a nonabelian simple loop $X$. If $\mathcal V$ is a variety of loops in which associativity propagates, $X$ is a finite loop in which associativity propagates and every subloop of $X$ is either nonabelian simple or contained in $\mathcal V$, then associativity propagates in $\mathbf{HSP}(X)\lor\mathcal V$. We study the propagating core $\mathcal S_{[x(yz)=(xy)z]}$ of Steiner loops with respect to associativity. While this is not a variety, we exhibit many varieties contained in $\mathcal S_{[x(yz)=(xy)z]}$, each providing a solution to Rajah's problem, i.e., a variety of loops not contained in Moufang loops in which Moufang Theorem holds.

math.GR↗

arXiv2020

Idempotent solutions of the Yang-Baxter equation and twisted group division

Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups $(X,*)$ are obtained by twisting the left division operation in groups (that is, they are of the form $x*y=ψ(x^{-1}y)$ for a group $(X,\cdot)$ and its automorphism $ψ$), and they correspond to idempotent latin solutions. We solve the isomorphism problem for idempotent latin solutions.

math.QA↗

arXiv2019

Permutations in two dimensions that maximally separate neighbors

We characterize all permutations on even-by-even grids that maximally separate neighboring vertices. More precisely, let $n_1$, $n_2$ be positive even integers, let $I(n_1,n_2)=\{1,\dots,n_1\}\times\{1,\dots,n_2\}$ be the $n_1\times n_2$ grid, let $d$ be the $L_1$ metric on $I(n_1,n_2)$, and let $N=\{\{x,y\}\in I(n_1,n_2)\times I(n_1,n_2):d(x,y)=1\}$ be the set of neighbors in $I(n_1,n_2)$. We characterize all permutations $π$ of $I(n_1,n_2)$ that maximize $\sum_{\{x,y\}\in N} d(π(x),π(y))$.

math.CO↗

arXiv2019

Enumeration of racks and quandles up to isomorphism

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders $n\le 13$ up to isomorphism, improving upon the previously known results for $n\le 8$ and $n\le 9$, respectively. The enumeration is based on the classification of subgroups of small symmetric groups up to conjugation, on a representation of racks and quandles in symmetric groups due to Joyce and Blackburn, and on a number of theoretical and computational observations concerning the representation. We explicitly find representatives of isomorphism types of racks of order $\le 11$ and quandles of order $\le 12$. For the remaining orders we merely count the isomorphism types, relying in part on the enumeration of $2$-reductive racks and $2$-reductive quandles due to Jedlička, Pilitowska, Stanovský and Zamojska-Dzienio.

math.QA↗

arXiv2019

Bol loops and Bruck loops of order $pq$ up to isotopism

Let $p>q$ be odd primes. We classify Bol loops and Bruck loops of order $pq$ up to isotopism. When $q$ does not divide $p^2-1$, the only Bol loop (and hence the only Bruck loop) of order $pq$ is the cyclic group of order $pq$. When $q$ divides $p^2-1$, there are precisely $\lfloor(p-1+4q)(2q)^{-1}\rfloor$ Bol loops of order $pq$ up to isotopism, including a unique nonassociative Bruck loop of order $pq$.

math.GR↗